HP 48GII, User Manual

Page: 622 / 864

Page 18-51
Additional equations for linear regression
The summary statistics such as
Σ
x,
Σ
x
2
, etc., can be used to define the
following quantities:






− = ⋅ − = − =
∑ ∑ ∑
= = =
n
i
i
n
i
i x
n
i
i xx
x
n
x s n x x S
1 1
2
2
1
2
1
) 1 ( ) (
2
1 1
2
2
1
2
1
) 1 ( ) ( 





− = ⋅ − = − =
∑ ∑ ∑
= = =
n
i
i
n
i
i y
n
i
i y
y
n
y s n y y S












− = ⋅ − = − − =
∑ ∑ ∑ ∑
= = = =
n
i
i
n
i
i
n
i
i i xy
n
i
i i xy
y x
n
y x s n y y x x S
1 1 1 1
2
1
) 1 ( ) )( (
From which it follows that the standard deviations of x and y, and the
covariance of x,y are given, respectively, by
1 −
=
n
S
s xx
x
,
1 −
=
n
S
s
yy
y
, and
1 −
=
n
S
s
yx
xy
Also, the sample correlation coefficient is
.
yy xx
xy
xy
S S
S
r
⋅
=
In terms of

x,

y, S
xx
, S
yy
, and S
xy
, the solution to the normal equations is:
x b y a − =
,
2
x
xy
xx
xy
s
s
S
S
b = =
Prediction error
The regression curve of Y on x is defined as Y =
Α
+
Β⋅
x +
ε
. If we have a set
of n data points (x
i
, y
i
), then we can write Y
i
=
Α
+
Β⋅
x
i
+
ε
I
, (i = 1,2,…,n),
where Y
i
= independent, normally distributed random variables with mean
(
Α + Β⋅
x
i
) and the common variance
σ 2
;
ε
i
= independent, normally distributed
random variables with mean zero and the common variance
σ 2
.